5 not required 6 answer each of the following as true

5. Not required.6.Answer each of the following as true or false.(a) All vectors of the form (a,0,-a) form a subspace ofR3.(b) InRn,kcxk=ckxk.(c) Every set of vectors inR3containing two vectors is linearly inde-pendent.(d) The solution space of the homogeneous systemAx=0is spannedby the columns.(e) If the columns of ann×nmatrix form a basis forRn, so do therows.(f) IfAis an 8×8 matrix such that the homogeneous systemAx=0has only the trivial solution then rank(A)<8.(g) Not required.(h) Every linearly independent set of vectors inR3contains three vec-tors.(i) IfAis ann×nsymmetric matrix, then rank(A) =n.(j) Every set of vectors spanningR3contains at least three vectors.Solution.(a) True. Indeed, these vectors form exactly

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54CHAPTER 6.VECTOR SPACESspan{(1,0,-1)}which is a subspace (by Theorem 6.3, p. 285).(b) False. Just takec=-1 andx6=0. You contradict the fact thatthe length (norm) of any vector is≥0.(c) False.x= (1,1,1) andy= (2,2,2) are linearly dependent inR3because 2x-y=0.(d) False.The solution space of the homogeneous systemAx=0isspanned by the columns which correspond to the columns of the reducedrow echelon form which do not contain the leading ones.(e) True. In this case the column rank ofAisn. But then also the rowrank isnand so the rows form a basis.(f) False. Just look to Corollary 6.5, p. 335, forn= 8.(h) False. For instance, each nonzero vector alone inR3forms a linearlyindependent set of vectors.(i) False. For example, the zeron×nmatrix is symmetric, but has notthe rank =n(it has zero determinant).(j) True. The dimension of the subspace ofR3spanned by one or twovectors is≤2, but dimR3= 3.

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56CHAPTER 8.DIAGONALIZATION(expanding successively along the first column). The corresponding char-acteristic equation has the solutionsa11, a22, ..., ann.T4.Show thatAandAThave the same eigenvalues.Solution.Indeed, these two matrices have the same characteristic poly-nomials:det(λIn-AT) = det((λIn)T-AT) = det((λIn-A)T)Th.3.1=det(λIn-A).

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